Later, the study of topological phases was extended to higher order and the previous mentioned TIs now
belong to the class of the first order[1]}, [2]}, [3]}. One way to distinguish the TIs
belonging to different orders is through their edge states. For example, a Chern insulator, which is a
type of first-order TIs characterized by the Chern number or quantized Hall conductance, has gapless
chiral states at its edge. On the other hand, the edge states are still gapped for a second-order TI in
two dimensions. However, if the boundaries of the sample are compatible with the protecting symmetry,
then gapless states will exist at the corners. The robustness of these gapless corner states is
guaranteed by the second-order bulk topology.
